WASHINGTON — With a final accord in sight, the 12 nations negotiating a trans-Pacific trade agreement linking 40 percent of the global economy have set a last round of talks for late July on the remaining issues on the most ambitious trade deal in a generation.

Outstanding controversies include access to Canada’s agriculture market, Australian concerns over American pharmaceutical patent rules, Peru’s rain forest management, Chinese components in Vietnamese textile exports and labor organizing rights in Vietnam and Mexico. The dispute over access to Canada’s protected dairy and poultry markets is so fierce that some participants say they believe Canada could drop out of the talks.

But the prize remains a massive trade zone stretching from Canada and Chile to Australia and Japan. It would be a legacy-defining achievement for President Obama, who showed last month he was willing to forsake most of his party to achieve it. Mr. Obama was able to persuade only 41 of Congress’s 234 Democrats to vote to give him expanded trade-negotiating powers — or trade promotion authority — and Democrats now want him to make good on producing a deal more of them can support.

“They have a decision to make: Do they want to achieve broader bipartisan support than they got for T.P.A. or don’t they?” asked Representative Sander Levin of Michigan, ranking Democrat on the Ways and Means Committee and an observer at the talks.

United States officials feel confident enough a deal is at hand that they have scheduled a meeting among the chief negotiators at the Westin Maui Resort & Spa in Hawaii during the last four days in July and have notified Congress that they expect this to be the last one. Still, it will be some time before a deal is ratified. Under terms set by Congress in trade negotiating legislation last month, a July 31 agreement could not be signed until Oct. 31 or more likely the beginning of November. Congress cannot begin considering it until December.

Yet the Office of the United States Trade Representative is projecting confidence.

“It’s not just U.S., but a number of other countries are reasonably confident negotiations can be concluded at this ministerial,” said Daniel Price, a senior international economic adviser in George W. Bush’s White House, referring to the Hawaii talks.

The final push began on Sunday when Vice President Joseph R. Biden Jr. met with Prime Minister Stephen Harper of Canada before the Women’s World Cup final in Vancouver. On Monday, Nguyen Phu Trong, general secretary of Vietnam’s Communist Party, arrived in Washington to meet with the United States trade representative, Michael Froman, ahead of meetings on Tuesday with Mr. Obama, Mr. Biden and Secretary of State John Kerry.

Some of the most difficult disputes in the Trans-Pacific Partnership involve Vietnam: governance of its state-owned enterprises, the rights of labor unions independent of state control and Vietnam’s desire to lower or eliminate import tariffs on Vietnamese textiles made with inputs like yarn from China, a country that is not part of the talks.

On Tuesday, Wendy Cutler, the acting deputy trade representative, will travel to Japan to complete a bilateral agreement over access to Japan’s automotive and agricultural markets. People with knowledge of those talks say the Obama administration is satisfied with Japan’s offer on auto and auto parts exports, as well as pork, beef and wheat. But Japan’s continued protection of rice farming remains a sticking point.

Mr. Froman is expected to travel to Mexico at the end of the week, with more trips planned for next week.

The closeness of a deal has only amplified the delicacy and complexity of the talks. Democrats and unions are pushing Mr. Obama to hold a hard line on labor rights, especially in Vietnam and Mexico, but if those countries cannot be certified as in compliance with the agreement’s union rules, other parts of the deal cannot be put into effect, said Lori Wallach, a critic of the agreement, with Public Citizen’s Global Trade Watch. That has politicians in countries like Japan worried they will take a hit for signing an agreement that will not be enforced for years.

The United States is pressing Canada to open its agriculture market, but if Washington cuts its own deal with Ottawa, Australia and New Zealand could protest. Peru is questioning whether to accept politically difficult environmental protections for its forests if a promised increase to its exports is held hostage to Mexico’s anti-labor practices.

Australia and New Zealand have protested United States insistence linking the accord’s intellectual property protections with patent rules that could slow the adoption of generic drugs in those countries’ national health programs.

Mr. Obama promised repeatedly that he would deliver the most progressive trade accord in history, and “the time for rhetoric has passed,” Mr. Levin said.

Still, the progress is remarkable for an agreement that started in 2008 with just four countries and the United States interested in only a small part of the talks. Mr. Obama’s promised “pivot” in foreign affairs toward Asia gave the negotiations more momentum. Vietnam entered the talks in 2009, changing the tone as capitalist powers began negotiating rules of commerce with a Communist economy. Japan’s decision to join the talks in 2013 gave the proposed accord enormous economic heft.

“I know there are intense feelings in this country about globalization, but the reality is it’s here to stay. We have to shape it,” said Ron Kirk, Mr. Obama’s first trade representative, who pushed the president into the talks. “The underlying rationale for the United States being involved in these negotiations is still compelling.”

A version of this article appears in print on , on Page B1 of the New York edition with the headline: Trade Deal Negotiators See a Wrap in Late July. Order Reprints | Today’s Paper | Subscribe